# Co-algebra

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2010 Mathematics Subject Classification: Primary: 16T15 [MSN][ZBL]

A module $A$ over a commutative ring $k$ with two homomorphisms, "comultiplication" $\phi : A \to A \otimes_k A$ and "counit" $\epsilon : A \to k$ such that the diagrams $$\begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ {}^\phi\downarrow & \ & \downarrow{}^{1 \otimes \phi} \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array}$$ and $$\begin{array}{ccccc} A \otimes A & \stackrel{\phi}{\leftarrow} & A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ & & & & \\ & \searrow{}^{\epsilon\otimes1}\ & \Vert & {}^{1\otimes\epsilon}\swarrow & \\ & & & & \\ & & A & & \end{array}$$ are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring $k$.

Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. Hopf algebra).

How to Cite This Entry:
Co-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-algebra&oldid=40943
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article